Turning a P.D.E to an O.D.E

I have a question, if we define a (for example) parabolic P.D.E in a spatiotemporal surface (1D in space), and also its initial and boundary vlues, would it be possible to solve the equation not in the entire domain, but in a space-time path.

u-xx+c*u-t=0

u(x=0)=0, u(x=l)=d, u(t=0)=0

we limit the domain in

a*x+b*t=0

u-xx+c*u-(-a/b*x)=0

boundary condition??

therefore we can eliminate t from the main equation, and simply deal with the O.D.E. and with changing the a and b and also interpolation, find the a good approximation of the exact answer of the main P.D.E in the entire domain.

I do not know is this approach is correct or not, but it if is, the main problem would be modeling the boundary and initial condition in a the path-domain.

We project the answer in a 2-D somain (space and time) on a 1-D path.